Hi
You could try some motor lodges in places you would like to stay in. I would avoid hotels in the centre of Auckland, we stayed in one on Queen Street for a week when we arrived in Sept and were kept awake most nights by very loud disco music.

However, I see that the Internet connection is paid, and I have just sent them an email asking if any discounts are available for longer stay at the hotel and of the price and available discounts for the internet connection. Might be worth checking (I am also planning to go to Auckland once approved for PR and will need a place to stay while I scout to rent an apartment).

My work has booked us three weeks at the Quadrant in Auckland (right near my work in CBD). We're staying in a 1 bedroom and they also have 2 bedrooms available. They also have washers and dryers in the apartments (big plus for us since we're bringing a 6 month old)